1.

What is the synthesis equation of the discrete time signal x(n), whose Fourier transform is X(ω)?(a) \(2π\int_0^2π X(ω) e^jωn dω\)(b) \(\frac{1}{π} \int_0^{2π} X(ω) e^jωn dω\)(c) \(\frac{1}{2π} \int_0^{2π} X(ω) e^jωn dω\)(d) None of the mentionedThe question was posed to me by my school teacher while I was bunking the class.My doubt stems from Frequency Analysis of Discrete Time Signal topic in chapter Frequency Analysis of Signals and Systems of Digital Signal Processing

Answer»

Right answer is (c) \(\frac{1}{2π} \int_0^{2π} X(ω) e^jωn dω\)

For explanation I would say: We KNOW that the Fourier transform of the discrete time SIGNAL x(n) is

X(ω)=\(\sum_{n=-∞}^∞ x(n)e^{-jωn}\)

By calculating the inverse Fourier transform of the above equation, we get

x(n)=\(\frac{1}{2π} \int_0^{2π} X(ω) e^{jωn} dω\)

The above equation is KNOWN as SYNTHESIS equation or inverse transform equation.


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